Quantifiers and Negation

Question

What is the negation of the following statement?

∀ ∃! (((,)∨~(,))⟷(,))

I have a problem to get the negation of ∃! (unique existential quantification) in such that problem.

Answer 1

Begin by drilling down into the uniqueness quantifier $${\neg\forall x~\exists{!}y~S(x,y)\\\neg\forall x~\exists y~(S(x,y)\wedge\forall z~(S(x,z)\to y=z))}$$

Now just apply the usual rules of quantifier duality, and conditional negation.

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Since the statement to be negated is that: "Every first-term has exactly one second-term which satisfies $S( ,)$", therefore the negation should express that: "There is some first-term with either none or at least two second-terms that satisfy $S(,)~$ (ie if there is one such second-term, then there is at least another.)."

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Finally replace $S(x,y)$ with $((P(x,y)\vee \lnot Q(x,y))\leftrightarrow R(x,y))$